From a point A the angle of elevation of a mountain is 13.713.7degrees°​, and from a point​ B, directly behind​ A, the angle of elevation is 10.410.4​°. See the figure to the right. If the distance between A and B is 77 ​miles, approximate the height of the mountain in feet.

To solve this problem, we will use the concept of trigonometry, particularly the tangent function in right triangles, and make use of the angles of elevation given from points A and B. Here's the process:

### Step 1: Define the Variables

- Let \( h \) be the height of the mountain in miles.
- Let \( d \) be the horizontal distance from point A to the base of the mountain.
- The distance between A and B is given as \( 77 \) miles.
- The angle of elevation at point A is \( 13.7^\circ \).
- The angle of elevation at point B is \( 10.4^\circ \).

From these variables, we can form two right triangles, one at point A and one at point B.

### Step 2: Use the Tangent Function

From point A, the tangent of the angle of elevation \( \theta_A = 13.7^\circ \) is:

\[
\tan(13.7^\circ) = \frac{h}{d}
\]

From point B, which is \( 77 \) miles behind A, the angle of elevation \( \theta_B = 10.4^\circ \) gives:

\[
\tan(10.4^\circ) = \frac{h}{d + 77}
\]

### Step 3: Set up the System of Equations

We now have two equations:
1. \[ h = d \cdot \tan(13.7^\circ) \]
2. \[ h = (d + 77) \cdot \tan(10.4^\circ) \]

### Step 4: Solve the System of Equations

Substitute \( h = d \cdot \tan(13.7^\circ) \) from the first equation into the second equation:

\[
d \cdot \tan(13.7^\circ) = (d + 77) \cdot \tan(10.4^\circ)
\]

Now, we solve for \( d \). After that, we can use the value of \( d \) to find \( h \).

Let me compute that for you.The horizontal distance from point A to the base of the mountain is approximately \( 234.6 \) miles. The height of the mountain is approximately \( 301,959 \) feet.

Do you have any questions or would you like further details?

Here are 5 related questions to deepen your understanding:
1. How does the tangent function relate to the angle of elevation in trigonometry?
2. What is the significance of using different points (A and B) with different angles of elevation?
3. How would the solution change if the distance between A and B was increased or decreased?
4. What assumptions do we make about the mountain's shape in this type of problem?
5. How can we apply the same principles to find the height of buildings or other structures?

**Tip**: Always check if the angles of elevation are given in degrees or radians, as this affects the computation.

From a point A the angle of elevation of a mountain is 13.7°, and from a point B, directly behind A, the angle of elevation is 10.4°. If the distance between A and B is 77 miles, approximate the height of the mountain in feet.

Learn how to approximate the height of a mountain using trigonometry and angles of elevation. From points A and B, with angles of 13.7° and 10.4°, and a distance of 77 miles between them, solve this step-by-step problem to find the mountain's height in feet. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation